Lines Matching refs:bm
21 \frac{\partial \rho}{\partial t} + \nabla \cdot \bm{U} &= 0 \\
22 \frac{\partial \bm{U}}{\partial t} + \nabla \cdot \left( \frac{\bm{U} \otimes \bm{U}}{\rho} + P \bm…
23 … \nabla \cdot \left( \frac{(E + P)\bm{U}}{\rho} -\bm{u} \cdot \bm{\sigma} - k \nabla T \right) - \…
27 where $\bm{\sigma} = \mu(\nabla \bm{u} + (\nabla \bm{u})^T + \lambda (\nabla \cdot \bm{u})\bm{I}_3)…
28 …bm{U}=\rho \bm{u}$, where $\bm{u}$ is the vector velocity field), $E$ the total energy density (de…
31 P = \left( {c_p}/{c_v} -1\right) \left( E - {\bm{U}\cdot\bm{U}}/{(2 \rho)} \right) \, ,
39 \frac{\partial \bm{q}}{\partial t} + \nabla \cdot \bm{F}(\bm{q}) -S(\bm{q}) = 0 \, ,
45 \bm{q} = \begin{pmatrix} \rho \\ \bm{U} \equiv \rho \bm{ u }\\ …
52 \bm{F}(\bm{q}) &=
54 \bm{U}\\
55 {(\bm{U} \otimes \bm{U})}/{\rho} + P \bm{I}_3 \\
56 {(E + P)\bm{U}}/{\rho}
57 \end{pmatrix}}_{\bm F_{\text{adv}}} +
60 - \bm{\sigma} \\
61 - \bm{u} \cdot \bm{\sigma} - k \nabla T
62 \end{pmatrix}}_{\bm F_{\text{diff}}},\\
63 S(\bm{q}) &=
66 \rho \bm{b}\\
67 \rho \bm{b}\cdot \bm{u}
77 \bm{q}_N (\bm{x},t)^{(e)} = \sum_{k=1}^{P}\psi_k (\bm{x})\bm{q}_k^{(e)}
83 … we first multiply the strong form {eq}`eq-vector-ns` by a test function $\bm v \in H^1(\Omega)$ a…
86 …ega} \bm v \cdot \left(\frac{\partial \bm{q}_N}{\partial t} + \nabla \cdot \bm{F}(\bm{q}_N) - \bm{…
89 with $\mathcal{V}_p = \{ \bm v(\bm x) \in H^{1}(\Omega_e) \,|\, \bm v(\bm x_e(\bm X)) \in P_p(\bm{I…
95 \int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) \…
96 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}(\bm{q}_N)\,dV & \\
97 + \int_{\partial \Omega} \bm v \cdot \bm{F}(\bm q_N) \cdot \widehat{\bm{n}} \,dS
98 &= 0 \, , \; \forall \bm v \in \mathcal{V}_p \,,
102 where $\bm{F}(\bm q_N) \cdot \widehat{\bm{n}}$ is typically replaced with a boundary condition.
105 …bm v \!:\! \bm F$ represents contraction over both fields and spatial dimensions while a single do…
116 \bm{q}_N^{n+1} = \bm{q}_N^n + \Delta t \sum_{i=1}^{s} b_i k_i \, ,
123 k_1 &= f(t^n, \bm{q}_N^n)\\
124 k_2 &= f(t^n + c_2 \Delta t, \bm{q}_N^n + \Delta t (a_{21} k_1))\\
125 k_3 &= f(t^n + c_3 \Delta t, \bm{q}_N^n + \Delta t (a_{31} k_1 + a_{32} k_2))\\
127 k_i &= f\left(t^n + c_i \Delta t, \bm{q}_N^n + \Delta t \sum_{j=1}^s a_{ij} k_j \right)\\
134 f(t^n, \bm{q}_N^n) = - [\nabla \cdot \bm{F}(\bm{q}_N)]^n + [S(\bm{q}_N)]^n \, .
140 The implicit formulation solves nonlinear systems for $\bm q_N$:
143 \bm f(\bm q_N) \equiv \bm g(t^{n+1}, \bm{q}_N, \bm{\dot{q}}_N) = 0 \, ,
146 where the time derivative $\bm{\dot q}_N$ is defined by
149 \bm{\dot{q}}_N(\bm q_N) = \alpha \bm q_N + \bm z_N
152 …in terms of $\bm z_N$ from prior state and $\alpha > 0$, both of which depend on the specific time…
157 …\frac{\partial \bm f}{\partial \bm q_N} = \frac{\partial \bm g}{\partial \bm q_N} + \alpha \frac{\…
179 …\int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) …
180 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}(\bm{q}_N)\,dV & \\
181 + \int_{\partial \Omega} \bm v \cdot \bm{F}(\bm{q}_N) \cdot \widehat{\bm{n}} \,dS & \\
182 … \int_{\Omega} \nabla\bm v \tcolon\left(\frac{\partial \bm F_{\text{adv}}}{\partial \bm q}\right) …
183 \nabla \cdot \bm{F} \, (\bm{q}_N) - \bm{S}(\bm{q}_N) \right) \,dV &= 0
184 \, , \; \forall \bm v \in \mathcal{V}_p
196 …\int_{\Omega} \bm v \cdot \left( \frac{\partial \bm{q}_N}{\partial t} - \bm{S}(\bm{q}_N) \right) …
197 - \int_{\Omega} \nabla \bm v \!:\! \bm{F}(\bm{q}_N)\,dV & \\
198 + \int_{\partial \Omega} \bm v \cdot \bm{F}(\bm{q}_N) \cdot \widehat{\bm{n}} \,dS & \\
199 …\int_{\Omega} \nabla\bm v \tcolon\left(\frac{\partial \bm F_{\text{adv}}}{\partial \bm q}\right) \…
200 & = 0 \, , \; \forall \bm v \in \mathcal{V}_p
206 In both {eq}`eq-weak-vector-ns-su` and {eq}`eq-weak-vector-ns-supg`, $\bm\tau \in \mathbb R^{5\time…
207 …bm F_{\text{adv}}}{\partial \bm q}$ (rather than its transpose) can be explained via an ansatz for…
208 The forward variational form can be readily expressed by differentiating $\bm F_{\text{adv}}$ of {e…
212 \diff\bm F_{\text{adv}}(\diff\bm q; \bm q) &= \frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \d…
214 \diff\bm U \\
215 (\diff\bm U \otimes \bm U + \bm U \otimes \diff\bm U)/\rho - (\bm U \otimes \bm U)/\rho^2 \diff\rho…
216 (E + P)\diff\bm U/\rho + (\diff E + \diff P)\bm U/\rho - (E + P) \bm U/\rho^2 \diff\rho
223 :::{dropdown} Stabilization scale $\bm\tau$
224 A velocity vector $\bm u$ can be pulled back to the reference element as $\bm u_{\bm X} = \nabla_{\…
225 …oundary layer element of dimension $(1, \epsilon)$, for which $\nabla_{\bm x} \bm X = \bigl(\begin…
227 The ratio $\lVert \bm u \rVert / \lVert \bm u_{\bm X} \rVert$ is a covariant measure of (half) the …
228 …in the direction of a unit vector $\hat{\bm n}$ is given by $\lVert \bigl(\nabla_{\bm X} \bm x\big…
229 While $\nabla_{\bm X} \bm x$ is readily computable, its inverse $\nabla_{\bm x} \bm X$ is needed di…
232 The cell Péclet number is classically defined by $\mathrm{Pe}_h = \lVert \bm u \rVert h / (2 \kappa…
236 \mathrm{Pe} = \frac{\lVert \bm u \rVert^2}{\lVert \bm u_{\bm X} \rVert \kappa}.
242 \tau = \frac{\xi(\mathrm{Pe})}{\lVert \bm u_{\bm X} \rVert},
247 For advection-diffusion, $\bm F(q) = \bm u q$, and thus the SU stabilization term is
250 \nabla v \cdot \bm u \tau \bm u \cdot \nabla q = \nabla_{\bm X} v \cdot (\bm u_{\bm X} \tau \bm u_{…
266 \tau_c &= \frac{C_c \mathcal{F}}{8\rho \trace(\bm g)} \\
274 + \bm u \cdot (\bm u \cdot \bm g)\right]
275 + C_v \mu^2 \Vert \bm g \Vert_F ^2}
278 where $\bm g = \nabla_{\bm x} \bm{X}^T \cdot \nabla_{\bm x} \bm{X}$ is the metric tensor and $\Vert…
285 + \frac{\bm u \cdot (\bm u \cdot \bm g)}{C_a}
286 + \frac{\kappa^2 \Vert \bm g \Vert_F ^2}{C_d} \right]^{-1/2}
294 … c_{\tau} \frac{2 \xi(\mathrm{Pe})}{(\lambda_{\max \text{abs}})_i \lVert \nabla_{x_i} \bm X \rVert}
297 … for linear elements, $\hat{\bm n}_i$ is a unit vector in direction $i$, and $\nabla_{x_i} = \hat{…
298 The flux Jacobian $\frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \cdot \hat{\bm n}_i$ in each …
305 where $u_i = \bm u \cdot \hat{\bm n}_i$ is the velocity component in direction $i$ and $a = \sqrt{\…
310 \lambda_{\max \text{abs}} \Bigl( \frac{\partial \bm F_{\text{adv}}}{\partial \bm q} \cdot \hat{\bm …
341 …thcal{V}_p^\mathrm{parent} = \{ \bm v(\bm x) \in H^{1}(\Omega_e^\mathrm{parent}) \,|\, \bm v(\bm x…
356 \bm M u_N = \int_0^{L_x} \int_0^{L_y} u \psi^\mathrm{parent}_N \mathrm{d}y \mathrm{d}x
358 where $\bm M$ is the mass matrix for $\mathcal{V}_p^\mathrm{parent}$, $u_N$ the coefficients of the…
362 \bm M [\langle \phi \rangle_z]_N = \int_0^{L_x} \int_0^{L_y} \left [\frac{1}{L_z} \int_0^{L_z} \phi…
368 \bm M [\langle \phi \rangle_z]_N = \frac{1}{L_z} \int_\Omega \phi(x,y,z,t) \psi^\mathrm{parent}_N(x…
388 \bm M [\langle \phi \rangle]_N = \frac{1}{L_z + (T_f - T_0)} \int_\Omega \int_{T_0}^{T_f} \phi(x,y,…
434 \overline{\phi} - \nabla \cdot (\beta (\bm{D}\bm{\Delta})^2 \nabla \overline{\phi} ) = \phi
437 …ution field, $\bm{\Delta} \in \mathbb{R}^{3 \times 3}$ a symmetric positive-definite rank 2 tensor…
441 \int_\Omega \left( v \overline \phi + \beta \nabla v \cdot (\bm{D}\bm{\Delta})^2 \nabla \overline \…
442 - \cancel{\int_{\partial \Omega} \beta v \nabla \overline \phi \cdot (\bm{D}\bm{\Delta})^2 \bm{\hat…
446 … from integration-by-parts is crossed out, as we assume that $(\bm{D}\bm{\Delta})^2 = \bm{0} \Left…
449 For homogenous filtering, $\bm{\Delta}$ is defined as the identity matrix.
457 B(\Delta; \bm{r}) =
459 1 & \Vert \bm{r} \Vert \leq \Delta/2 \\
460 0 & \Vert \bm{r} \Vert > \Delta/2
465 For inhomogeneous anisotropic filtering, we use the finite element grid itself to define $\bm{\Delt…
468 … most conveniently defined by $\bm{\Delta} = \bm{g}^{-1/2}$ where $\bm g = \nabla_{\bm x} \bm{X} \…
470 #### Filter width scaling tensor, $\bm{D}$
471 The filter width tensor $\bm{\Delta}$, be it defined from grid based sources or just the homogenous…
473 The definition for $\bm{D}$ then becomes
476 \bm{D} =
484 In the case of $\bm{\Delta}$ being defined as homogenous, $\bm{D}\bm{\Delta}$ means that $\bm{D}$ e…
498 …the wall parallel directions to be no less than the original filter width defined by $\bm{\Delta}$.
501 Under these assumptions, $\bm{D}$ then becomes:
504 \bm{D} =
513 While we define $\bm{D}\bm{\Delta}$ to be of a certain physical filter width, the actual width of t…
525 \frac{\partial E}{\partial t} + \nabla \cdot (\bm{u} E ) - \kappa \nabla E = 0 \, ,
528 with $\bm{u}$ the vector velocity field and $\kappa$ the diffusion coefficient.
534 …We have solved {eq}`eq-advection` applying zero energy density $E$, and no-flux for $\bm{u}$ on th…
543 …nflow}} \bm v \cdot \bm{F}(\bm q_N) \cdot \widehat{\bm{n}} \,dS = \int_{\partial \Omega_{inflow}} …
550 …{outflow}} \bm v \cdot \bm{F}(\bm q_N) \cdot \widehat{\bm{n}} \,dS = \int_{\partial \Omega_{outflo…
561 \frac{\partial \rho}{\partial t} + \nabla \cdot \bm{U} &= 0 \\
562 \frac{\partial \bm{U}}{\partial t} + \nabla \cdot \left( \frac{\bm{U} \otimes \bm{U}}{\rho} + P \bm…
563 \frac{\partial E}{\partial t} + \nabla \cdot \left( \frac{(E + P)\bm{U}}{\rho} \right) &= 0 \, , \\
567 …rho= 1$ (Specific Gas Constant, $R$, is 1), and $\bm{u}=(u_1,u_2,0)$ while the perturbation $\delt…
585 \int_{\Omega} \nu_{SHOCK} \nabla \bm v \!:\! \nabla \bm q dV
600 …adient unit vector. This density gradient unit vector is defined as $\hat{\bm j} = \frac{\nabla \r…
603 h_{SHOCK} = 2 \left( C_{YZB} \,|\, \bm p \,|\, \right)^{-1}
624 \bm{U} &= \bm U_\infty \\
625 …\frac{-(\bar{x}^2 + \bar{y}^2)}{2\sigma^2}\right)\right) + \frac{\bm U_\infty \cdot \bm U_\infty}{…
650 Given the force components $\bm F = (F_x, F_y, F_z)$ and surface area $S = \pi D L_z$ where $L_z$ i…
664 …on is defined in terms of the Exner pressure, $\pi(\bm{x},t)$, and potential temperature, $\theta(…
667 … c_p - c_v)\theta(\bm{x},t)} \pi(\bm{x},t)^{\frac{c_v}{ c_p - c_v}} \, , \\ e &= c_v \theta(\bm{x}…
671 For this problem, we have used no-slip and non-penetration boundary conditions for $\bm{u}$, and no…
721 \bm{u}(\bm{x}, t) = \bm{\overline{u}}(\bm{x}) + \bm{C}(\bm{x}) \cdot \bm{v}'
726 \bm{v}' &= 2 \sqrt{3/2} \sum^N_{n=1} \sqrt{q^n(\bm{x})} \bm{\sigma}^n \cos(\kappa^n \bm{d}^n \cdot …
727 \bm{\hat{x}}^n &= \left[(x - U_0 t)\max(2\kappa_{\min}/\kappa^n, 0.1) , y, z \right]^T
731 Here, we define the number of wavemodes $N$, set of random numbers $ \{\bm{\sigma}^n,
732 \bm{d}^n, \phi^n\}_{n=1}^N$, the Cholesky decomposition of the Reynolds stress
733 tensor $\bm{C}$ (such that $\bm{R} = \bm{CC}^T$ ), bulk velocity $U_0$,
735 0.5 \min_{\bm{x}} (\kappa_e)$.
829 $\varepsilon$, $C_{ij}$, and $\overline{\bm u}$) or by calculation (for $q^n$).
842 The `STGRand.dat` file is the table of the random number set, $\{\bm{\sigma}^n,
843 \bm{d}^n, \phi^n\}_{n=1}^N$. It has the format:
852 | Math | Label | $f(\bm{x})$? | $f(n)$? |
854 | $ \{\bm{\sigma}^n, \bm{d}^n, \phi^n\}_{n=1}^N$ | RN Set | No | Yes |
855 | $\bm{\overline{u}}$ | ubar | Yes | No |
859 | $\bm{R}$ | R_ij | Yes | No |
860 | $\bm{C}$ | C_ij | Yes | No |
873 S(\bm{q}) = -\sigma(\bm{x})\left.\frac{\partial \bm{q}}{\partial \bm{Y}}\right\rvert_{\bm{q}} \bm{Y…
876 where $\bm{Y}' = [P - P_\mathrm{ref}, \bm{0}, 0]^T$, and $\sigma(\bm{x})$ is a linear ramp starting…
877 … a pressure-primitive anomaly $\bm Y'$ converted to conservative source using $\partial \bm{q}/\pa…